#7826541

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#43118	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,32)(30,31);(1,2)(3,7)(4,8)(5,6)(17,19)(18,23)(20,29)(21,31)(22,28)(24,25)(26,27)(30,32);(9,10,12,14)(11,13,16,15)(19,28)(21,29)(23,27)(25,32);(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,29)(21,28)(22,31)(23,25)(26,30)(27,32);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,30)(21,25)(22,26)(24,31)(27,28)(29,32);(1,4,5,3)(2,8,6,7)(17,21,22,29)(18,25,26,32)(19,20,28,31)(23,30,27,24)

Polycyclic presentation:

2^9|c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7733737);